Abstract
We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker’s relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics.
Highlights
The propagation of nonlinear waves in a network is an important topic
For large amplitude shallow water waves our numerical calculations show that the energy entering a branch can vary from 20% to 50% depending on the symmetry of the fork
Before considering the nonlinear shallow water equations, we analyze the simpler case of a class of scalar 2D nonlinear wave equations
Summary
The propagation of nonlinear waves in a network is an important topic. As an example, consider a hydrological network which is prone to floods. For large amplitude shallow water waves our numerical calculations show that the energy entering a branch can vary from 20% to 50% depending on the symmetry of the fork These studies point out the importance of the angle. We revisit the problem of shallow water propagation in 2D forks using our homothetic reduction procedure to obtain approximate conservation laws and compare them with the numerical solutions. This is a first formal justification of Stoker’s interface conditions This angle independent reduction holds for a general class of scalar nonlinear wave equations, for example the 2D sine-Gordon equation or the 2D reaction-diffusion equation; it confirms the results of [2].
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